Darwin W. Smith

Extension of Koopmans’ theorem. I. Derivation

It is shown that the spin orbitals that diagonalize the generalized Hartree−Fock potentials [Int. J. Quantum Chem. Symp., 8, 501 (1974)] for a correlated wavefunction, together with the corresponding orbital energies, give a natural extension of Koopmans’ theorem to correlated reference states.

Extension of Koopmans’ theorem. II. Accurate ionization energies from correlated wavefunctions for closed‐shell atoms

Extended Koopmans’ theorem ionization energies are presented for MC−SCF wavefunctions of the 1s2 2s2 (1S) ground state isoelectronic series, Li− through O4+, and also for some larger CI wavefunctions of Be and B+. Correlation in these reference states reduces the error in the Koopmans’ ionization energy for the 2s electron to approximately 0.01−0.08 eV (1/100−1/20 the SCF Koopmans’ error) in all states except the negative ion, for which the MC−SCF extended Koopmans’ error of 0.25 eV was of the same magnitude but of opposite sign to the SCF error. Our extended Koopmans’ energies for 1s ionization were only slightly better than the corresponding SCF values. Koopmans’ theorem does not yield a 2p ionization energy (1s2 2s2 → 1s2 2p), but the extended Koopmans’ theorem yields an ionization energy whose error is about one−third the SCF 2s error.

Natural Orbitals and Geminals of the Beryllium Atom

The first‐ and second‐order density matrices for Watsons 37‐configuration beryllium wavefunction have been studied. Particular attention has been given to the form of the second‐order density matrix and the way it changes in passing from a Hartree—Fock to a separated electron pair to a configuration‐interaction wavefunction. The effect of intershell correlation in beryllium is interpreted as causing a forced delocalization of the natural geminals, and a splitting of the singlet—triplet degeneracy which would otherwise be present in the geminal occupation numbers.The first‐ and second‐order density matrices for Watsons 37‐configuration beryllium wavefunction have been studied. Particular attention has been given to the form of the second‐order density matrix and the way it changes in passing from a Hartree—Fock to a separated electron pair to a configuration‐interaction wavefunction. The effect of intershell correlation in beryllium is interpreted as causing a forced delocalization of the natural geminals, and a splitting of the singlet—triplet degeneracy which would otherwise be present in the geminal occupation numbers.

Lower‐Bound Procedure for Energy Eigenvalues by the Partitioning Technique

A lower‐bound procedure for obtaining energy eigenvalues by use of the partitioning technique and bracketing theorem, which have been developed by Lowdin, is extended to the case of a multidimensional reference manifold and is applied to the ground state of the two‐electron isoelectronic series. Except for H−, the agreement between upper and lower bounds is quite satisfactory. The process of obtaining lower bounds for excited states is considered.

Asymptotic behavior of atomic Hartree–Fock orbitals

The restricted Hartree–Fock equations are solved at large r for closed shell atomic states. Except for states containing only s orbitals, the Hartree–Fock orbitals obey the formula, φi(r)∼kiYlimi(ϑ,φ) rβ−λ−1 exp(−ζr), where −1/2ζ2=eho, the highest occupied orbital energy, where β= (nuclear charge – number of electrons +1−ζ)/ζ, where λ+1=0 for the highest occupied orbital, where λ=‖ li−lho ‖ if li≠lho, where λ=2 if li=lho≠0 and i≠ho, where λ=2lmin+1 if li=lho, and where lmin is the smallest nonzero l value of the occupied Hartree–Fock orbitals.

Extension of Koopmans’ theorem. IV. Ionization potentials from correlated wavefunctions for molecular fluorine

Uncorrelated (Hartree–Fock) ab initio calculations have proven unable to predict the energy ordering of 3σg and 1πu molecular orbitals as observed in electron spectroscopy experiments on fluorine (F2). The correct ordering was obtained, however, by applying an extension of Koopmans’ theorem [J. Chem. Phys. 62, 113 (1975)] to an MC–SCF correlated wavefunction for F2 which contained only one configuration beyond the one used in Hartree–Fock. With the addition of still further configurations to the wavefunction for the neutral molecule, the ordering of the extended Koopmans’ valence orbital energies was maintained and the correspondence improved between those and the experimental values.

Density Matrix Study of Atomic Ground and Excited States. I. Beryllium Ground State

Various configuration‐interaction wavefunctions for the Be ground state built up from Slater‐type orbitals are reported. Their 1‐ and 2‐matrices are examined in detail and compared with those previously reported in the literature. It is shown how the quality of a basis set and the omission of just a few configurations noticeably alter the density matrix structure. Further, it is pointed out that the occupation numbers of some important natural states are in general quite sensitive to the goodness of the wavefunction. In the choice of configurations to be included in a truncated CI expansion over an extended basis set by performing stepwise full CI expansions over small basis sets, the role played by singly excited configurations is emphasized. The use of density matrices as a global representation of the wavefunction is also stressed.

Lower bounds to energy eigenvalues for the Stark effect in a rigid rotator.

Upper and lower bounds have been calculated for the energy levels of a rigid rotator in an electric field, in order to study the problems associated with the use of the partitioning method for bracketing an eigenvalue of the Schrodinger equation. Results of arbitrarily high accuracy are possible in this example.

N‐Representability Problem for Fermion Density Matrices. I. The Second‐Order Density Matrix with N =3

The N representability problem is the problem of how to recognize whether a given pth‐order reduced density matrix is derivable from an antisymmetaic N‐particle wavefunction ψ by integrating out all but coordinates from ψ*ψ.The conditions under which a given second‐order density matrix is N representable by a three‐particle wavefunction are discussed.

Density Matrix Study of Atomic Ground and Excited States. II. Beryllium 1S Excited State

An 85‐configuration CI wavefunction for the first beryllium 1S excited state is reported and discussed. The total energy obtained is −14.404518 a.u. compared to the Hartree‐Fock and experimental nonrelativistic energies of −14.3622 and −14.4178 a.u., respectively. The 1 and 2 matrices and geminal energies of the Be ground and first excited states are compared in detail to clarify the changes undergone by the natural states on excitation.